33 problems

Integer-spectrum conjecture. The algebraic eigenvectors span the three lowest levels of the spin chain. When $M\teq1$, the fourth algebraic energy, $\mathsf{E}=3$, is the fifth low…

For the even-site ground state $|\Psi^{(2n)}\rangle$, define … and set $\alpha_i=(1+\sigma_i^z)/2$. Korepin–Izergin–Baxter ferromagnetic-string formula. The probabilities of format…

For the even-site ground state $|\Psi^{(2n)}\rangle$, define the normalized expectation of an operator $O$ by … and define $\alpha_i=(1+\sigma_i^z)/2$. The twisted XXZ next-nearest…

Let $|\Psi^{(2n)}\rangle$ be the ground-state vector for the twisted XXZ chain with $2n$ sites, let $W^{(2n)}$ index its components, and let $A_n$ be the number of alternating-sign…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, let … and let $A_n$ be the number of alternating-sign $n\times n$ matrices…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, and let $A_n$ denote the number of alternating-sign $n\times n$ matrices.…

Let $H^{(N)}$ be the XXZ Hamiltonian with twisted boundary conditions, with $N$ an even number of sites, twisting angle $\phi$, anisotropy $\Delta$, and total spin $z$-axis project…

Transverse matrix-element conjecture. The first-order splitting matrix element satisfies

Let $N=2n+1$ be odd, and let $a_1,\ldots,a_k$ denote the local observables defining the ferromagnetic-string correlations. Ferromagnetic-string probability conjecture. The correlat…

Let $N=2n+1$ be odd, and let $a_j$ denote the local observable used for the spin-chain correlations. Correlation formula conjecture. The next-nearest-neighbor correlation is … The…

Let $N=2n+1$ be odd, let ${\mathcal N}_n$ be the norm of the ground state, and let the state vector have components in the spin-configuration basis. Sum-of-components conjecture. T…

Norm formula conjecture. The norm is

Let $H$ be the Hamiltonian referred to as ${\rm (H1)}$, with an odd number $N$ of sites, and let $E$ and $S_z$ denote an energy eigenvalue and the total spin in the $z$-direction.…

Completeness conjecture. The Bethe Ansatz completely determines the spectrum for all but finitely many values of $\Delta$ with $|\Delta|<\varepsilon$.

Global velocity-properties conjecture. The conclusions of Proposition 1 hold for all

Let $N$ be the length of the spin-one chain and let $x$ be its anisotropy parameter. Consider the Hamiltonian with anti-diagonal twist. Anti-diagonal-twist conjecture. For every…

Let $N$ be the length of the spin-one chain and let $x$ be its anisotropy parameter. Consider the Hamiltonian with diagonal twist and its eigenvalue degeneracies. Odd-length diagon…

Let $N$ be the length of the spin-one chain and let $x$ be its anisotropy parameter. Consider the Hamiltonian with diagonal twist and its spectrum, including eigenvalue degeneracie…

Let $s$ be the spin, $N$ the system size, and $ell$ the number of Bethe roots. A solution is called regular if it is a regular solution of the Bethe ansatz equations, physical sing…

For odd $N=2n+1$, consider the cyclic spin-chain Hamiltonian with parameters $\lambda,\zeta$ and its three translation-invariant zero-energy singlets $\Psi_N^a$, indexed by…

Let $H_N$ be the Fateev–Zamolodchikov spin-chain Hamiltonian, with twist angle $\phi$ and total third spin component $S_N^3$. The elliptic zero-mode conjecture. For odd $N=2n+1$, t…

In the trigonometric case $y=0$, choose the singlet components to be coprime polynomials in $x$ and normalise their leading terms as specified in the source. Let $N$ be written as…

For twist angle $\phi=\pi$, write the supersymmetry singlet as $|\Psi\rangle=\sum_\mu\psi_\mu|\mu\rangle$, with the components chosen as coprime polynomials in $x,y$. The norm-fact…

Let $H_N(x,y)$ be the Hamiltonian on $N$ sites, with twist angle $\phi$, and let $S_N^3$ denote the total third spin component. The representative singlet conjecture. For every $N$…

Let $H_N(x,y)$ be the spin-chain Hamiltonian with twist angle $\phi$, depending on couplings $x$ and $y$. The spectral exchange conjecture. For twist angle $\phi=\pi$, the spectrum…